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Investigations on Joule Heating Applications by Multiphysical Continuum Simulations in Nanoscale Systems

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Investigations on Joule Heating Applications by Multiphysical Continuum Simulations in Nanoscale Systems Manuel Feuchter INVESTIGATIONS ON JOULE HEATING APPLICATIONS BY MULTIPHYSICAL CONTINUUM SIMULATIONS IN NANOSCALE SYSTEMS SCHRIFTENREIHE DES INSTITUTS FÜR ANGEWANDTE MATERIALIEN BAND 43 Investigations on Joule heating applications in nanoscale systems M. FEUCHTER 43 Manuel Feuchter Investigations on Joule heating applications by multiphysical continuum simulations in nanoscale systems Schriftenreihe des Instituts für Angewandte Materialien Band 43 Karlsruher Institut für Technologie (KIT) Institut für Angewandte Materialien (IAM) Eine Übersicht aller bisher in dieser Schriftenreihe erschienenen Bände finden Sie am Ende des Buches. Investigations on Joule heating applications by multiphysical continuum simulations in nanoscale systems by Manuel Feuchter Dissertation, Karlsruher Institut für Technologie (KIT) Fakultät für Maschinenbau Tag der mündlichen Prüfung: 08.07.2014 Impressum Karlsruher Institut für Technologie (KIT) KIT Scientific Publishing Straße am Forum 2 D-76131 Karlsruhe KIT Scientific Publishing is a registered trademark of Karlsruhe Institute of Technology. Reprint using the book cover is not allowed. www.ksp.kit.edu This document – excluding the cover – is licensed under the Creative Commons Attribution-Share Alike 3.0 DE License (CC BY-SA 3.0 DE): http://creativecommons.org/licenses/by-sa/3.0/de/ The cover page is licensed under the Creative Commons Attribution-No Derivatives 3.0 DE License (CC BY-ND 3.0 DE): http://creativecommons.org/licenses/by-nd/3.0/de/ Print on Demand 2014 ISSN 2192-9963 ISBN 978-3-7315-0261-6 DOI 10.5445/KSP/1000042982 Investigations on Joule heating applications by multiphysical continuum simulations in nanoscale systems Zur Erlangung des akademischen Grades Doktor der Ingenieurwissenschaften der Fakultat¨ fur¨ Maschinenbau Karlsruher Institut fur¨ Technologie (KIT) genehmigte DISSERTATION von Dipl.–Ing. Manuel Klaus Ludwig Feuchter geboren am 20.11.1983 in Wertheim Tag der mundlichen¨ Prufung:¨ 08.07.2014 Hauptreferent: Prof. Dr.–Ing. M. Kamlah Korreferent: Prof. Dr. rer. nat. C. Jooss Korreferent: Prof. Dr. rer. nat. O. Kraft ‘No subject has more extensive relations with the progress of industry and the natural sciences; for the action of heat is always present, it influences the processes of the arts, and occurs in all the phenomena of the universe.’ – Jean Baptiste Joseph Fourier [cf. 118, p. 241] Abstract A requirement for future thermoelectric applications are poor heat conducting materials. Nowadays, several nanoscale approaches are used to decrease the thermal conductivity of this material class. A promising approach applies thin multilayer structures, composed of known materials. Here, the nanoscale stacking affects the heat conduction and provokes new emergent thermal properties; However, the heat conduction through these materials is not well understood yet. Therefore, a reliable measurement technique is required to measure and understand the heat propagation through these materials. In this work, the so-called 3!-method is focused upon to investigate thin strontium titanate (STO) and praseodymium calcium manganite (PCMO) layer materials. Previously unexamined macroscopic influence factors within a 3!-measurement are considered in this thesis by Finite Element simulations. Thus, this work furthers the overall understanding of a 3!-measurement, and allows precise thermal conductivity determinations. Moreover, new measuring configurations are developed to determine isotropic and anisotropic thermal conductivities of samples from the micro- to nanoscale. Since no analytic solutions are available for these configurations, a new evaluation methodology is presented to determine emergent thermal conductivities by Finite Element simulations and Neural Networks. v Kurzfassung Zukunftige¨ thermoelektrische Anwendungen erfordern thermisch schlecht leitende Materialien. Hierfur¨ werden heutzutage verschiedene Ansatze¨ verfolgt um die Warmeleitf¨ ahigkeitszahl¨ dieser Materialklasse zu verringern. Ein vielversprechender Ansatz verwendet dunne¨ Vielschichtstrukturen die sich aus bekannten Materialien zusammen setzen. Hier wird durch nanoskaliges Schichten die Warmeleitung¨ beeinflußt und es werden neue thermische Eigenschaften hervorgerufen. Jedoch ist die Warmeleitung¨ durch solch ein Material bis heute noch nicht bis ins Detail verstanden. Zu diesem Zweck bedarf es einer verlaßlichen¨ Messmethode, um die Warmeausbreitung¨ durch solch ein Material messen und somit verstehen zu konnen.¨ Diese Arbeit konzentriert sich auf die so genannte 3!-Methode zur Erforschung dunner¨ Strontium Titanat (STO) und Praseodym Calcium Manganit (PCMO) Schichtmaterialien. Bisher unbeachtete makroskopische Einflußfaktoren, die wahrend¨ einer 3!-Messung auftreten, werden in dieser Arbeit durch Finite Element Simulationen berucksichtigt.¨ Dadurch tragt¨ dieses Werk zum Gesamtverstandnis¨ einer 3!-Messung bei und erlaubt eine prazise¨ Bestimmung der Warmeleitf¨ ahigkeitszahl.¨ Daruber¨ hinaus werden neue Messkonfigurationen zur Bestimmung der isotropen als auch anisotropen Warmeleitf¨ ahigkeitszahl¨ fur¨ mikro- bis nanoskalige Proben entwickelt. Da fur¨ diese Messkonfigurationen keine analytischen Losungen¨ verfugbar¨ sind, wird eine neue Methodik zur Auswertung und Bestimmung der Warmeleitf¨ ahigkeitszahl¨ vorgestellt, die Finite Element Simulationen und Neuronale Netze kombiniert. vii Contents Nomenclature xiii 1 Introduction 1 1.1 Thermoelectricity and material selection . 8 1.2 Heat conduction in solids . 16 2 The 3!-method 21 2.1 Concept, measurement principle and geometry configurations . 21 2.2 Top down geometry . 26 2.2.1 Heat source on bulk materials . 26 2.2.2 Heat source on layer-substrate materials . 35 2.3 Bottom electrode geometry . 40 2.3.1 Heat source between bulk-like materials . 40 2.3.2 Heater substrate platform . 42 3 Finite Element Model 45 3.1 Governing equations . 46 3.1.1 Temperature field . 46 3.1.2 Electromagnetic field . 52 3.1.3 Mechanical field . 59 3.2 Transient analysis . 63 3.3 Eigenfrequency analysis . 65 3.4 Mesh-block building system . 67 ix 4 Investigation of varying structures and multiphysical couplings 69 4.1 Top down geometry . 69 4.1.1 Heat source on bulk materials . 69 4.1.1.1 Validation of Finite Element Model . 69 4.1.1.2 Real substrate size . 71 4.1.1.3 Geometrical variations of the heater and its material properties . 73 4.1.1.4 General temperature distribution for a heater substrate system . 76 4.1.1.5 Temperature dependent resistivity of the heater . 78 4.1.1.6 Three-dimensional heat conduction in the substrate . 80 4.1.1.7 Radiation at the heater’s surface . 84 4.1.1.8 Surface roughness . 86 4.1.1.9 Skin effect in the heater . 88 4.1.1.10 Thermal expansion and stresses . 90 4.1.1.11 Eigenfrequency analysis . 94 4.1.2 Heat source on layer-substrate materials . 96 4.1.2.1 Monolayer configurations . 97 4.1.2.2 Multilayer configurations . 104 4.2 Bottom electrode geometry . 107 4.2.1 Bulk-like materials . 107 4.2.2 Thin layers . 111 4.2.3 Application of heat sink . 114 4.2.4 Patterned multilayer structures . 119 4.3 Membrane structures . 121 4.3.1 Two-dimensional models . 122 x 4.3.2 Three-dimensional 6-pad heater structure . 127 4.4 Pillar and pad structure . 138 5 Methodology to determine the isotropic and anisotropic thermal conductivity 155 5.1 General methodology . 155 5.2 Neural Network and Inverse Problem . 157 6 Application examples 163 6.1 STO layer on an STO substrate . 163 6.2 Bottom electrode geometry . 168 7 Summary 175 A Detailed information on formula 181 A.1 Modulated voltage . 181 A.2 Information about the approximate solution . 182 A.3 Definition of the skin depth and derivation of decoupled equations for the magnetic and electric field intensity .. 185 A.4 Derivation for the complex Helmholtz-Equation . 188 B Material properties and constants 189 C Technical drawing of the 6-pad heater structure 191 Publications 193 References 197 Acknowledgment - Danksagung 221 xi Nomenclature Greek Symbols α linear temperature coefficient [ 1=K] β linear thermal expansion coefficient [ 1=K] ΩH area of the heater Ωlay/m area of the layer or material Ωs area of the substrate σs stress tensor Υ amplitude of motion " infinitesimal strain tensor "T thermal strain χ, $; ; φ complex terms of Borca-Tasciuc’s solution ∆R electric resistance oscillation [ Ω ] B ∆TH temperature amplitude due to Borca-Tasciuc including heater properties [K] B ∆Th temperature amplitude due to Borca-Tasciuc without heater properties [K] C ∆Th temperature amplitude due to Cahill [K] C ∆Tlay temperature amplitude due to layer [K] ∆Tˇ general temperature oscillation [K] ∆T temperature amplitude in the heater/material [ K ] permittivity [ As=Vm ] 0 vacuum constant permittivity [ As=Vm ] relative permittivity of specific material [ ] r − η Neural Network gradient scaling parameter xiii Γ boundary for the partial differential equation γ Euler-Maschoneri constant [ ] − ΓH1 4 boundary for the magnetic field − ΓInt interface boundary κeff effective thermal conductivity [W=mK ] κ thermal conductivity [W=mK ] κe thermal conductivity due to electrons [ W=mK ] κH thermal conductivity of the heater [W=mK ] κk; κk+1 thermal conductivities of two contacting materials [W=mK ] κlay thermal conductivity of the layer [W=mK ] κmcrs cross-plane thermal conductivity of the investigated material [W=mK ] κmin in-plane thermal conductivity of the investigated material [W=mK ] κm thermal conductivity of the investigated material [ W=mK ] κph thermal conductivity due to phonons [ W=mK ] κs thermal conductivity of the substrate [W=mK ] ( ) Neural
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